Posted
by
samzenpuson Wednesday April 25, 2012 @10:56PM
from the learning-to-count dept.

An anonymous reader writes "The Yupno people of New Guinea have provided clues to the origins of the number-line concept, and suggest that the familiar concept of time may be cultural as well. From the article: 'Tape measures. Rulers. Graphs. The gas gauge in your car, and the icon on your favorite digital device showing battery power. The number line and its cousins – notations that map numbers onto space and often represent magnitude – are everywhere. Most adults in industrialized societies are so fluent at using the concept, we hardly think about it. We don't stop to wonder: Is it 'natural'? Is it cultural? Now, challenging a mainstream scholarly position that the number-line concept is innate, a study suggests it is learned."

I don't get your comment. I teach math to six year olds once a week. They "get" the number line, in that they use it as a useful tool for calculation, and can understand how numbers equate to divisions on the paper. Is it innate? Probably not. Is it something that many six year olds in the US culture have? From my experience, yes.

Where the article veers into the absurd is the suggestion that we should consider "bringing the human saga" into teaching math, and that math isn't objective fact, or black and white. Math is freaking math. There is right and wrong, black and white.

I teach math to six year olds once a week. They "get" the number line, in that they use it as a useful tool for calculation, and can understand how numbers equate to divisions on the paper. Is it innate? Probably not. Is it something that many six year olds in the US culture have? From my experience, yes.

When my kids started school, they had to be taught how to use number lines, number grids for multiplication, how to divide by 2 and so on, just as much as they had to be taught how to read. None of it is innate, as far as I can see.

Watch kids play with Lego sometime. They'll be able to tell you why their sibling has the very brick they were going to use to make their creation. Number line they get. Fungibility of Lego bricks, they don't.

Math is a set of ways of mapping some of the real world into a world of artificial symbols and concepts.

No. The science which maps real-world phenomena onto artificial symbols and concepts is known as physics. Mathematics is only concerned with the artificial symbols and concepts, independent of whether they can be mapped to real-world entities (many things cannot).

"The map is not the territory"

Yes. And mathematics is about the map and its rules, without caring about the territory, or even if it corresponds to a territory at all. If you want to learn about the territory, use physics. And yes, you'll use maps (i.e. mathematics) there, too. But those maps are not arbitrary, but carefully adapted to the territory as far as we know it, and actively developed to improve how well it maps the territory.

So in the map/territory picture you have:

Mathematics: The science of maps. Doesn't care about what the maps mean, or if they mean anything at all. As long as a map is consistent, it is accepted as valid map.

Physics: The science of territory. Uses maps to describe the territory. A map is considered valid only if it describes the relevant aspects of the mapped territory sufficiently well.

Similarly, human linguistics generally concerns itself only with mappings between symbolic concepts with no thought as to how those are truly internally represented nor how synthesis into external representation occurs.

No it most certainly does not. That's "semiotics". "Linguistics" is a much broader field.

Your ten year old probably doesn't understand the number line. Sure, he can put a few numbers on a line, but ask him to put a million, and a thousand on the line. Try it yourself, you may be surprised.

In order to save time, paper, and ink, I made my number line logarithmic.

I hope you made two of them for the synergy effect.

Slide rule joke of the day:When Noah told his menagerie "go forth and multiply", two snakes replied: "We can't, we're adders!"Noah then built a wooden table, placed the snakes on it, and much joy and spawn ensued.Because on a log table, even adders can multiply.

If your 10 year old doesn't ALREADY understand the number line, you have failed. Hell, if your 6 year old doesn't understand it, you've failed.

Then I guess I failed. My seven year old son is at the top of his 2nd grade class in math. Be he was doing the number line exercise in Khan Academy [khanacademy.org] about two weeks ago, and he needed some help. Once I explained the concept, and gave him a few examples, he "got it", and was able to do the exercises. But it was not intuitive. He needed an explanation.

Well, numbers are abstract. I'm not sure how a number line representation, which can take real shape would be an intuitive extension of an artificial concept. It isn't. Actually, it's the other way around, I would think. The number lines help us understand numbers and it's numbers that aren't intuitive.

I wonder how far this goes! Is the notion of the counting numbers innate? I have heard that monkeys cannot count beyond 4. The way that people figured this out is that if five hunters go into a forest as a group, split up and hide. Then one by one, four hunters leave one at a time. The fifth hunter stays in hiding, the monkeys come out of hunting, and the hunter shoots a monkey. This does not happen when there are less than five hunters initially.

The way that people figured this out is that if five hunters go into a forest as a group, split up and hide. Then one by one, four hunters leave one at a time. The fifth hunter stays in hiding, the monkeys come out of hunting, and the hunter shoots a monkey. This does not happen when there are less than five hunters initially.I should hope not: if there are four hunters initially, then one by one four hunters leave, there are no hunters left to shoot the monkey. And if there are 3 or fewer hunters initially than the scenario's impossible.

Thats not necessarily even counting on the monkeys behalf. A lot of neuroscientists reckon we can process about 4 separate things in our mind simultaneously , and then use a variety of clever tricks to work around it (Ie counting!) and if that stretches across species. So conciably the monkeys are just at their limit of how many dudes they can track at once, rather than an inability to count beyond 4.

Numbers are not an intuitive concept. As I've learned more and more math, I've had numerous discussions about this topic. The conclusions that tend to be reached are that sets are intuitive. A set is very intuitive, it's just a bunch of objects that are grouped together. You may not THINK of these things as sets, but that's what they are. You have a pile of apples, or a herd of sheep, or a group of hunters. Those are all sets of objects (or some philosophers would argue that there's a difference between the set and the group of physical objects, but I don't think that this ruins the intuition here). You can also label those things however you want, or not label them at all. Very intuitive. But numbers are when intuition starts to get messed up. A number can be disassociated from a concrete set, and that can make it hard to deal with, if you're not used to it. What is 1? What does it mean? What does it even mean to talk about 1 sheep, if it's completely hypothetical? There's no concrete sheep there, so what does it MEAN to be talking about 1 sheep? It's not even like you're talking about a sheep that's going to be born, or that belongs to your neighbors. This sheep is basically just imaginary. That's really a huge jump in cognition, especially when you start to consider other crazy things about numbers, like what's the biggest number, and what's a negative number, and what if you can't divide your numbers evenly. Anyways, nothing scholarly to back this up, just my experience in mathematics:)

The problem with this argument is that it assumes that set THEORY is intuitive, which I do not agree with. While a SET is an intuitive concept, the ZF axioms of set theory and what they imply are NOT intuitive. There may be basic operations that are more intuitive, like the union of two sets or the intersection of two sets, but that intuition is almost entirely tied to the physical manifestation of the set. As soon as you introduce the formal idea of a set, especially as an abstract construct, I believe that, just like what I said about numbers, you remove a large amount of the basic intuition behind them. While a lot of the things that happen here seem intuitive to us, I feel like that is almost solely due to the fact that we are introduced to this abstraction at such an early age, and we deal with it so much, that we internalize it. Without that exposure, I'm not so sure the abstractions of sets and numbers is totally intuitive.

You can tell it was supposed to be the Peano construction (and not something else) because the GP defined zero as the empty set and 2 as {0,1}. The error was to also define 2 as {{{}}}, which is clearly not equivalent to {0,1} (since the former set has cardinality 1 and the latter has cardinality 2).

This is an incredibly common mistake even for math undergrads and good evidence that set theory really isn't very intuitive. There's a reason New Math failed.

No, not joking. There already have been studies that show different cultures have different counting systems. For example, many cultures will have only the most basic of numbers (1, 2, 3, 4, 5) and then jump into the "many" category. Another example of the non-intuitive nature of numbers? 0. That one took a while to catch on. Third example? Describe to me a forest with -10 trees or a person with -1 apple. Negative numbers were not intuitive either. Notice I am avoiding those wonderful numbers like fractions, irrational numbers (pi, e, the square root of two, etc), and complex numbers (i, the square root of -1... graph that on your number line!) - all of which are not intuitive in and of themselves. Final example? If numbers are intuitive, why does it take so long to teach our young to count? Why do so few people understand the concept of billions and trillions of dollars of debt, or the vast distances of the universe, or the very tiny number which represents the time in which million/billion/trillions of molecules collide and interact when undergoing an exothermic reaction?

No, while you have been educated and indoctrinated into a system of numbers, that does not mean it is intuitive. Or another way to think of it - take the pro basketball player who has taught his muscles how to shoot a 3-pointer... he might argue that it is intuitive, meanwhile someone like me (who couldn't make a freaking free-throw shot) would say that it is definitely not intuitive.

I wonder how far this goes! Is the notion of the counting numbers innate?Counting exact numbers is not innate. There are some cultures that don't have words for an exact number beyond 3. That doesn't mean they don't understand quantities, just that they can't name a specific amount. It'd be like if somone showed you a thousand of something, and 1100 of something. You'd know the 1100 was more, but you wouldn't be certain by exactly how much more.

I'm inclined not to believe your oversimplification. I remember elementary school math, with whole chapters devoted to teaching the number line. Concepts such as greater/less, constant distance, visual estimation, and numberless comparisons are, or were, part of what gets taught in a school setting.

If you don't have the concept of a number line already, is it really that intuitive to stack 1 cup on top of another and consider it a measurement rather than an amount? Stacking things and coming up with a ruler based on that stacking seem like they are fairly distinct concepts, that one won't lead to the other.

If you read the article, you'll see that the subjects of the study do understand order, but that they lack the intuition of another property of the number line that you are so accustomed to that you're not aware of it. When asked to place numbers from 1 to 10 in order, control subjects (from the US) produce an arrangement like this:

1...2...3...4...5...6...7...8...9...10

The people of the Yupno Valley tend to do something more like this:

1.2.3.4...................5.6.7.8.9.10

A number line has more than order; it also has equal spacing. That idea seems not to be innate.

I'm not sure if they've fixed it yet, but the defaults for line charts in MS Excel were insanely set to have equal spacing between data points on one axis no matter what values they have.Thus you could have an axis that looked like:1 4 7 8 14 35IMHO that sort of defeats the purpose of a line graph. I can userstand linear or log scales but a random changing scale is pointless.

You would probably quite enjoy Noam Chomsky's latest work, The Science of Language. In it, he claims nothing is innate except the concept of Merge. Basically, it is only set theory and construction/deconstruction based upon that. Counting numbers is not innate; it is consequential of a certain kind of indoctrination. All humans can potentially do it, but it is not something inborn. Likewise, all humans can learn a spoken/written/signed language, but it is not inborn.

I haven't read the book, but what about subitizing [wikipedia.org], i.e. the ability to "perceive" a small number of items? If a three-week old baby can subitize up to three objects, I'd say that's an inborn ability.

Do you intuitively know what a continent is? If you said yes, post a reply then check out What are Continents? [youtube.com] - then post another reply to that.

As to the measuring cup example: if a number line is so intuitive to a measuring cups, why are so many sets of unmarked 1/4 cup, 1/3 cup, 1/2 cup, and 1 cup measuring cups sold? After all, shouldn't anyone just need a 1 cup measuring cup? For that matter, why need tablespoons and teaspoons? After all, a tablespoon is merely 1/16 of a cup and a teaspoon is 1/48

I don't have the reference to hand but I recall there is a South American tribe which don't have words for left and right as most languages do. There words are equivalent to "Up Valley" and "Down Valley"
Similarly, if I recall correctly, there's a Native American language that uses before and behind as an analog for time but the other way around to most languages. Their analogy is that you know the past and you can see what it in front of you so forward = the past. You can't see behind you and you don't know the future so behind = the future

The Piraha are in South America and they have a language that is lacking many words considered normal in other cultures. http://en.wikipedia.org/wiki/Pirah%C3%A3_language [wikipedia.org]. They give directions primarily in terms of the relation to the river (towards or away from the river or up or down the river) which may be what you are thinking of. There's a highly readable book about the tribe and their language- "Don't Sleep, There Are Snakes" by Daniel Everett, a linguist who spent decades with them. However, there's some degree of question by other scholars about how accurate Everett's description of their language was, and research is ongoing.

In Lingala (Kingshasa area in Congo), they only have one word which both means "yesterday" and "tomorrow". Basically things happen today or they happen not-today. This kind of makes sense in a climate that has no cold and hot season, and where it is useless (or even a very bad idea) to do typical northern stuff like plan way ahead, conserve food or make warm clothes. Most pre-Columbus south american indians saw time as a strictly circular thing, with everything always comming back.

FTA:
"In their time study with the Yupno, now in press at the journal Cognition, Nunez and colleagues find that the Yupno don't use their bodies as reference points for time – but rather their valley's slope and terrain. Analysis of their gestures suggests they co-locate the present with themselves, as do all previously studied groups. (Picture for a moment how you probably point down at the ground when you talk about "now.") But, regardless of which way they are facing at the moment, the Yupno point

Similarly, if I recall correctly, there's a Native American language that uses before and behind as an analog for time but the other way around to most languages. Their analogy is that you know the past and you can see what it in front of you so forward = the past. You can't see behind you and you don't know the future so behind = the future

Yes, that may well have been...

the Aymara of the Andes [seem to do the reverse, placing the past in front and the future behind]

Figuring out what isn't intuitive isn't useful, unless we also know what is. Pie graphs for gas gauges, showing the shrinkage of the tank fractionally? Or a circle in a circle shrinking within the "full" one?

"Also, we document that precise number concepts can exist independently of linear or other metric-driven spatial representations."

But TFA doesn't mention any of them, or what we could change a gas gauge to to be intuitive.

Perhaps one day they can figure out why my mother compulsively fills up once the gauge goes under 1/2, but my sister runs cars to empty on a regular basis, usually filling up only after the "e" is lit, sometimes long after.

Once a significant percentage of the population becomes interested in measuring pieces of land for various purposes, people will start associating numbers to lines.Because the amount of food is proportional to the surface of your land, and then... I personally feel it's quite natural, in this context, to associate numbers to geometrical constructs.

The same subject has been covered in "Here's looking to Euclid". It describes tests done on an Amazon tribe to see how they visually interpret numbers. Unlike most modern adults who visualize number spaced linearly, they visualized them spaced logarithmically. Their reasoning was that the intervals between numbers start (relatively) large and become smaller as the numbers get larger. i.e. from 1 to 2 it's a 100% increase but from 2 to 3 it's only a 33% increase and so on.

I imagine that a thickness gauge (which is what is *really* intuitive in the measuring-cup example) or a color-gauge would be more intuitive. The critical point here is that thicker is "more" and thinner is "less". Even with colors you can have "more red" or "less red". Numbers are a higher-form thought process. When dealing with a line system, your general intention is to gauge this same "more or less" comparions, but is abstracted through numbers which is based on a complex thought process of reading and

In the original task, people are shown a line and are asked to place numbers onto the line according to their size, with "1" going on the left endpoint and "10" (or sometimes "100" or "1000") going on the right endpoint.

Go to a class of college students in america, ask them to mark 10, 1 million, and 1 billion on a line, and 99% of them will draw 1 million closer to 1 billion. Usually a lot closer.

I read the article, and it wasn't clear to me what these people have discovered. Maybe I'll have to read the actual study. Or maybe anthropologists are better at understanding primitive cultures than their own.

I don't know why this result is surprising. I thought it was generally understand that counting (there are 10 sheep) and measurement (this fence is 10 feet long) were distinct concepts. The point of the number line is to establish a relationship between the two concepts.

Come to think of it, it should be obvious that a number line relates two distinct concepts, just from the form they usually take. A number line, with its regularly spaced markings perpendicular to the main line, has a form similar to that of a line graph, which shows a relationship between two distinct variables.

Oddly enough, I was telling my girlfriend just tonight that I'm not very visual, and tend to approach concepts best through symbols (numbers, words, etc.) I've always found graphical representations of math more-or-less useless (although they are cool sometimes) and prefer my math without the diagrams. She told me that I'm deeply weird.:)

My little brother was having problems with vector math. So, I threw together a vector visualiser in my game engine, and illustrated basic vector primitives, and operations. Within 15 minutes of moving them around on the screen and seeing the values and vectors change he understood normalising, and dot and cross products, as well as trigonomic primitives like sine and cosine, and tangent. I showed him how dot products are used to cull faces in games, and in lighting equations, and how cross products make

Neither is reading. Human beings evolved to see "in the round" and not in focused linear scans. When we were children, both my sister and I went through periods when we were just learning to write where we wrote everything "exactly" backwards, like a mirror image. And, it wasn't all the time. We both outgrew it very quickly, but I'm sure it's been studied by some -ologist out there.

Logical or not, the number line is equivalent to a finite list of axioms (field axioms, look 'em up, maybe with some stuff I forget atm). When we accept the truth of those axioms, all at once, then we begin studying 'the number line'.

Personally, studying unintuitive concepts via the language of mathematics interests me. That's how mathematics allows you to expand the list of things that you find intuitive. First, only the abstract language of mathematics describes some logical object. The logical object

Decades, months, and days of the week all have specific shapes, locations, and colors. They have always been the same as far as I can remember. Numbers you would use in calculating things have color, albeit past 10 they group in 10s. That is all the 20s are a yellow orange color, 30s purple blue, and so on. The personality of numbers is entirely about if they are prime or have prime factors or are odd. It's a simple good and bad type thing. 3 and 7 are sinister, 9 more so, 21 also. All are odd and ar

I once took a course in "Math philosophy" (a simple introduction course, with e.g. Gödel numbers, introduction to infinity, and things like that), and at the end of that course we were asked to write about something. I decided to ask friends about how they viewed numbers. To my surprise, everyone had pretty much their own unique way. I think I asked about 10 people. Some viewed numbers as colors ("the number 2 is of course blue" or something along that line), some viewed the numbers as on a traditional line, one guy thought of the numbers as being in a circle and you took one out as you wanted to use it and then had to put it back. Not everyone included the number zero (or negative numbers) in their explanation. My self, I see the natural numbers on a line, but the line has "angles" at the numbers 10 and 20. Perhaps this is because in my native language, the spoken words for 10..19 are not constructed in the same simple manner as 30..39, 40..49, and so on.

And how many thought in binary? Although I don't count every day in binary (the indoctrination into the decimal system is almost impossible to avoid in the Western world), I often catch myself finding binary patterns and thinking about things in a binary way (and if someone asks me to remember a number, the best way is to try to calculate its binary expression - the calculation and the resulting string fix into my memory a lot easier). Hell, when I run out of fingers counting in decimal, it's easier for m

Fascinating. When I was a kid numbers used to talk and fight with each other. Some numbers were good and some were bad. Not sure that's a very useful way to think of numbers because I am horrid at arithmetic.

'Cultural' is natural for us humans, so it is a daft question. A better question would be to ask whether this is something we are most likely to have learned through our early experience - and how. And I think the answer is likely to be that we learn the idea of "moreness" being a continuous thing from observing varying amounts of things - water in a glass etc, or the length of a piece of string; these concepts are clearly learned as and when you learn the words to describe them - ie. it is 'cultural'.

But many - maybe most - animals have the ability to gauge the relative size of things, and some, like the corvids - even seem able to count. Thus that would count as a 'natural' ability, I suppose.

The case with the Yupno seems to be that measurements aren't needed in their culture; one can muse over where that need arises from - it could be a result of trade, perhaps?

I read the article pointed to in the summary (which is a summary of the scholarly article). The study authors seem to have confused the idea that finding a single population that behaves this way (not arranging piles of oranges linearly along a line according to the number of oranges in a pile) with determining true innate human behavior. Find another dozen isolated groups, and then maybe. Find groups that have been only recently isolated and it will be more impressive.

If you grew up with the metric system you might not realize that common measurements used to be based on supposedly common items, so you had measurements dealing with what a man could hold with his arms around it, and the length of the King's erect cock or whatever. It's a natural advance to go from measuring things in terms of a fingertip to so many fingertip-units. I imagine it would have started with measuring distance, but it could as easily have been someone figuring it out by volume, this container holds so many of that container. Or this stick rolls over x times when it passes down the side of this object.

Fixed measurements, such as a number line or the 'natural numbers' offer a poor model of reality. Comparing apples to apples; few are equal. Some are bigger, more bruised, less ripe, more bitter.Hardly anything could be more alien than Euclidean space - we live on a mottled sphere. Straight lines are very much the exception.While convenient, 'intuitive' or 'natural' are hardly the best way to describe abstract shortcuts.